# Some very basic questions on topology!

#### Mathelogician

##### Member
hello everybody!
My knowledge in the field is so basic. So please answer as clear and simple as possible:
The following 4 questions are the ones i couldn't solve after trying!
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1-Prove [0,1] is not isometric to [0,2].
2-For which intervals [a,b] in R is the intersection of [a,b] and Q a clopen (closed and open) subset of the metric space Q?
3-Find a metric space in which the boundary of Mr(p)={xEM :d(x,p)< r} is not equal to the sphere of radius r at p ,{xEM : d(x,p)=r}.
4- For a subspace N of a metric space M,prove that if the openness of S c N is equivalent to openness of S in M, then N is open in M.
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The next is about my proof of the following problem:

Let (An) be a nested decreasing sequence of non-empty closed sets in the metric space M.
If M is complete(every Cauchy is convergent) and diam(An) -> 0 as n -> infinity, show that S=Intersection of (An) is exactly one point( note that diam X= sup{d(x,y): x,y E X} ).
My proof : (I use Euclidean metric on R for simplicity of the proof- it can be done for any other metric on R)
Lets x,y E S. we must show that d(x,y)=0.
Lets d(x,y)= t > 0. By the red part: there exists a N1 such that for all n>N1 , we have diam(An)<t
Let k > n. So diam(Ak)<t.
In other hand, d(x,y) =< diam(Ak)< t. So d(x,y)< t which contradicts the blue.

Now my question is : What was the usage of supposing Ai's as Closed subspaces of M and M's being complete, in this proof?

#### Opalg

##### MHB Oldtimer
Staff member
Re: Some very basic quastions on topology!

hello everybody!
My knowledge in the field is so basic. So please answer as clear and simple as possible:
The following 4 questions are the ones i couldn't solve after trying!
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

1-Prove [0,1] is not isometric to [0,2]. The interval [0,2] contains two points whose distance apart is 2, but the interval [0,1] does not.
2-For which intervals [a,b] in R is the intersection of [a,b] and Q a clopen (closed and open) subset of the metric space Q? Think about what difference it makes whether a and b are rational or irrational.
3-Find a metric space in which the boundary of Mr(p)={xEM :d(x,p)< r} is not equal to the sphere of radius r at p ,{xEM : d(x,p)=r}. Think about a space with a discrete metric.
4- For a subspace N of a metric space M, prove that if the openness of S c N is equivalent to openness of S in M, then N is open in M. What happens if you take S = N?
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(See the hints in red.)

The next is about my proof of the following problem:
Let (An) be a nested decreasing sequence of non-empty closed sets in the metric space M.
If M is complete(every Cauchy is convergent) and diam(An) -> 0 as n -> infinity, show that S=Intersection of (An) is exactly one point( note that diam X= sup{d(x,y): x,y E X} ).
My proof : (I use Euclidean metric on R for simplicity of the proof- it can be done for any other metric on R)
Lets x,y E S. we must show that d(x,y)=0.
Lets d(x,y)= t > 0. By the red part: there exists a N1 such that for all n>N1 , we have diam(An)<t
Let k > n. So diam(Ak)<t.
In other hand, d(x,y) =< diam(Ak)< t. So d(x,y)< t which contradicts the blue.

Now my question is : What was the usage of supposing Ai's as Closed subspaces of M and M's being complete, in this proof?
Your proof shows that S cannot contain more than one point. But how do you know that it contains any points at all? Answer: You need those extra conditions to ensure that S is not empty.